Transcript CSC 480: Artificial Intelligence

ARTIFICIAL INTELLIGENCE
Games
GAMES AND COMPUTERS

Games offer concrete or abstract competitions
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
“I’m better than you!”
Some games are amenable to computer treatment
mostly mental activities
 well-formulated rules and operators
 accessible state
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
Others are not
emphasis on physical activities
 rules and operators open to interpretation
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state not (easily or fully) accessible
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DEFINING COMPUTER GAMES
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Video game / computer game
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“a mental contest, played with a computer according to certain
rules for amusement, recreation, or winning a stake” [Zyda,
2005]
Serious game
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a mental contest, played with a computer in accordance with
specific rules, that uses entertainment to further training or
education [modified from Zyda, 2005]
[Zyda, 2005] Michael Zyda, “From Visual Simulation to Virtual Reality to Games,”
IEEE Computer, vol. 39, no. 9, pp. 25-32.
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WHAT ARE AND WHY STUDY GAMES?
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Games are a form of multi-agent environment
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What do other agents do and how do they affect our
success?
Cooperative vs. competitive multi-agent environments.
Competitive multi-agent environments give rise to
adversarial search a.k.a. games
Why study games?
Fun
 Interesting subject of study because they are hard

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RELATION OF GAMES TO SEARCH
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Search – no adversary
 Solution is (heuristic) method for finding goal
 Heuristics and CSP techniques can find optimal solution
 Evaluation function: estimate of cost from start to goal
through given nodes
 Examples: path planning, scheduling activities
Games – adversary
 Solution is strategy (strategy specifies move for every
possible opponent reply).
 Time limits force an approximate solution
 Evaluation function: evaluate “goodness” of
game position
 Examples: chess, checkers, Othello, backgammon
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ASPECTS OF COMPUTER GAMES
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Story
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defines the context and content of the game
Art
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presentation of the game to the user
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Software
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emphasis on visual display, sound
implementation of the game on a computer
Purpose
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entertainment
training
education
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EXAMPLES
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EXAMPLES
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GAME ANALYSIS
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Often deterministic
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the outcome of actions is known
sometimes an element of chance is part of the game
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Two-player, turn-taking
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one move for each player
Zero-sum utility function
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e.g. dice
what one player wins, the other must lose
Often perfect information
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fully observable, everything is known to both players about
the state of the environment (game)
not for all games
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e.g. card games with “private” or “hidden” cards
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TYPES OF GAMES
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GAMES AS ADVERSARIAL SEARCH
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Many games can be formulated as search problems
The zero-sum utility function leads to an adversarial
situation
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in order for one agent to win, the other necessarily has to
lose
Factors complicating the search task
potentially huge search spaces
 elements of chance
 multi-person games, teams
 time limits

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SINGLE-PERSON GAME
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Conventional search problem
identify a sequence of moves that leads to a winning state
 examples: Solitaire and Rubik’s cube
 little attention in AI
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Some games can be quite challenging
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some versions of solitaire
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CONTINGENCY PROBLEM
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Uncertainty due to the moves and motivations of the
opponent
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tries to make the game as difficult as possible for the
player
attempts to maximize its own, and thus minimize the
player’s utility function value
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TWO-PERSON GAME
 Games
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often called MIN and MAX
usually MAX moves first, then they take turns
 MAX
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wants a strategy to find a winning state
no matter what MIN does
 MIN
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with two opposing players
does the same
or at least tries to prevent MAX from winning
 Perfect
Decisions
traverse all relevant parts of the search tree
 often impractical because of time and space limitations
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EXAMPLE TRIVIAL GAME
 Deal
four playing cards out, face up
 Player 1 chooses one, player 2 chooses one
 Player 1 chooses another, player 2 chooses another
 And the winner is….
 Add the cards up
 The player with the highest even number
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Scores that amount
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ENTIRE SEARCH SPACE
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MOVING THE SCORES FROM
THE BOTTOM TO THE TOP
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MOVING A SCORE
WHEN THERE’S A CHOICE
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Use minimax
assumption
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CHOOSING THE BEST MOVE
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MINIMAX STRATEGY
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Optimal strategy for MAX
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generate the whole game tree
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calculate the value of each terminal state
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starting from the leaf nodes up to the root
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MAX selects the value with the highest node
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MAX assumes that MIN in its move will select
the node that minimizes the value
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assumes that both players play optimally
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MINIMAX PROPERTIES
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Based on depth-first
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Time complexity is O(bm)
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recursive implementation
exponential in the number of moves
Space complexity is O(bm)
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where b is the branching factor, m the maximum depth of
the search tree
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MINIMAX EXAMPLE
4
7
9
6
9
8
8
5
6
7
5
2
3
2
5
4
terminal nodes: values calculated from the utility function
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9
3
MINIMAX EXAMPLE
4
7
6
2
6
3
4
5
1
2
5
4
1
2
6
3
4
3
Min
4
7
9
6
9
8
8
5
6
7
5
2
3
2
5
4
other nodes: values calculated via minimax algorithm
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9
3
MINIMAX EXAMPLE
7
6
5
5
6
4
Max
4
7
6
2
6
3
4
5
1
2
5
4
1
2
6
3
4
3
Min
4
7
9
6
9
8
8
5
6
7
5
2
3
2
5
4
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9
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MINIMAX EXAMPLE
5
7
3
6
4
5
Min
5
6
4
Max
4
7
6
2
6
3
4
5
1
2
5
4
1
2
6
3
4
3
Min
4
7
9
6
9
8
8
5
6
7
5
2
3
2
5
4
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MINIMAX EXAMPLE
5
Max
5
7
3
6
4
5
Min
5
6
4
Max
4
7
6
2
6
3
4
5
1
2
5
4
1
2
6
3
4
3
Min
4
7
9
6
9
8
8
5
6
7
5
2
3
2
5
4
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9
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MINIMAX EXAMPLE
5
Max
5
7
3
6
4
5
Min
5
6
4
Max
4
7
6
2
6
3
4
5
1
2
5
4
1
2
6
3
4
3
Min
4
7
9
6
9
8
8
5
6
7
5
2
3
2
5
4
moves by Max and countermoves by Min
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9
3
IMPERFECT DECISIONS
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Complete search is impractical for most games
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Alternative: search the tree only to a certain depth
requires a cutoff-test to determine where to stop
 uses a heuristics-based evaluation function to estimate the
expected utility of the game from the leave nodes
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PRUNING
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Discards parts of the search tree
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guaranteed not to contain good moves
guarantee that the solution is not in that branch or sub-tree
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if both players make optimal decisions, they will never end up in that
part of the search tree
Results in substantial time and space savings
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as a consequence, longer sequences of moves can be explored
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ALPHA-BETA PRUNING
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Certain moves are not considered
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won’t result in a better evaluation value
Applies to moves by both players
a indicates the best choice for Max so far never decreases
 b indicates the best choice for Min so far never increases
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Extension of the minimax approach
results in the same move as minimax, but with less overhead
 prunes uninteresting parts of the search tree
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ALPHA-BETA EXAMPLE 1
[-∞, +∞]
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Max
[-∞, +∞]
a best choice for Max
b best choice for Min
Min
?
?
we assume a depth-first, left-to-right search as basic strategy
 the range of the possible values for each node are indicated
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initially [-∞, +∞]
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ALPHA-BETA EXAMPLE 2
[-∞, +∞]
5
Max
[-∞, 7]
Min
7
a best choice for Max
b best choice for Min

?
7
Min obtains the first value from a successor node
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ALPHA-BETA EXAMPLE 3
[-∞, +∞]
5
Max
[-∞, 6]
7
Min
6
a best choice for Max
b best choice for Min

?
6
Min obtains the second value from a successor node
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ALPHA-BETA EXAMPLE 4
[5, +∞]
5
7
6
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Max
Min
5
a best choice for Max
b best choice for Min
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5
5
5
Min obtains the third value from a successor node
this is the last value from this sub-tree, and the exact value is
known
Max now has a value for its first successor node, but hopes that
something better might still come
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ALPHA-BETA EXAMPLE 5
[5, +∞]
6
5
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Min
3
a best choice for Max
b best choice for Min

Max
[-∞, 3]
5
7
5
5
3
Min continues with the next sub-tree, and gets a better value
Max has a better choice from its perspective, however, and will
not consider a move in the sub-tree currently explored by Min

initially [-∞, +∞]
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ALPHA-BETA EXAMPLE 6
[5, +∞]
[-∞, 3]
5
7
6
5
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Max
Min
3
a best choice for Max
b best choice for Min
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5
5
3
Min knows that Max won’t consider a move to this sub-tree, and
abandons it
this is a case of pruning, indicated by
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ALPHA-BETA EXAMPLE 7
[5, +∞]
[-∞, 3]
5
7
6
5
3
a best choice for Max
b best choice for Min
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5
Max
[-∞, 6]
Min
6
5
3
Min explores the next sub-tree, and finds a value that is worse
than the other nodes at this level
if Min is not able to find something lower, then Max will choose
this branch, so Min must explore more successor nodes
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ALPHA-BETA EXAMPLE 8
[5, +∞]
[-∞, 3]
5
7
6
5
3
a best choice for Max
b best choice for Min
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5
Max
[-∞, 5]
6
Min
5
5
3
Min is lucky, and finds a value that is the same as the current
worst value at this level
Max can choose this branch, or the other branch with the same
value
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ALPHA-BETA EXAMPLE 9
5
[-∞, 3]
5
7
6
5
[-∞, 5]
6
Min
5
5
3
Min could continue searching this sub-tree to see if there is a
value that is less than the current worst alternative in order to
give Max as few choices as possible
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Max
3
a best choice for Max
b best choice for Min
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5
this depends on the specific implementation
Max knows the best value for its sub-tree
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ALPHA-BETA EXAMPLE OVERVIEW
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5
7
6
<= 3
5
3
a best choice for Max
b best choice for Min

Max
6
<=5
4
6
Min
5
4
5
7 -> 6 -> 5 -> 3
some branches can be pruned because they would never be
considered

after looking at one branch, Max already knows that they will not be
of interest since Min would choose a value that is less than what Max
already has at its disposal
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